# Terminology

The following terms, symbols, and decorators are used in text and diagrams throughout this guide.

## Notation

- Bold face variables indicate vectors or matrices and non-bold face variables represent scalars.
- The default frame for each variable is the local frame: $\ell{}$. Right [superscripts](#superscripts) represent the coordinate frame. If no right superscript is present, then the default frame $\ell{}$ is assumed. An exception is given by Rotation Matrices, where the lower right subscripts indicates the current frame and the right superscripts the target frame.
- Variables and subscripts can share the same letter, but they always have different meaning.

## Acronyms

| Acronym     | Expansion                                                                                                                                                                      |
| ----------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| AOA         | Angle Of Attack. Also named *alpha*.                                                                                                                                           |
| AOS         | Angle Of Sideslip. Also named *beta*.                                                                                                                                          |
| FRD         | Coordinate system where the X-axis is pointing towards the Front of the vehicle, the Y-axis is pointing Right and the Z-axis is pointing Down, completing the right-hand rule. |
| FW          | Fixed-Wing.                                                                                                                                                                    |
| MC          | MultiCopter.                                                                                                                                                                   |
| MPC or MCPC | MultiCopter Position Controller. MPC is also used for Model Predictive Control.                                                                                                |
| NED         | Coordinate system where the X-axis is pointing towards the true North, the Y-axis is pointing East and the Z-axis is pointing Down, completing the right-hand rule.            |
| PID         | Controller with Proportional, Integral and Derivative actions.                                                                                                                 |


## Symbols

| Variable                               | Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| -------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $x,y,z$                                | Translation along coordinate axis x,y and z respectively.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| $\boldsymbol{\mathrm{r}}$            | Position vector: $\boldsymbol{\mathrm{r}} = [x \quad y \quad z]^{T}$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| $$\boldsymbol{\mathrm{v}}$$          | Velocity vector: $\boldsymbol{\mathrm{v}} = \boldsymbol{\mathrm{\dot{r}}}$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| $$\boldsymbol{\mathrm{a}}$$          | Acceleration vector: $\boldsymbol{\mathrm{a}} = \boldsymbol{\mathrm{\dot{v}}} = \boldsymbol{\mathrm{\ddot{r}}}$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
| $$\alpha$$                            | Angle of attack (AOA).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| $$b$$                                  | Wing span (from tip to tip).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
| $$S$$                                  | Wing area.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
| $$AR$$                                 | Aspect ratio: $AR = b^2/S$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
| $$\beta$$                             | Angle of sideslip (AOS).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| $$c$$                                  | Wing chord length.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
| $$\delta$$                            | Aerodynamic control surface angular deflection. A positive deflection generates a negative moment.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
| $$\phi,\theta,\psi$$                | Euler angles roll (=Bank), pitch and yaw (=Heading).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| $$\Psi$$                              | Attitude vector: $\Psi = [\phi \quad \theta \quad \psi]^T$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
| $$X,Y,Z$$                              | Forces along coordinate axis x,y and z.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
| $$\boldsymbol{\mathrm{F}}$$          | Force vector: $\boldsymbol{\mathrm{F}}= [X \quad Y \quad Z]^T$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
| $$D$$                                  | Drag force.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| $$C$$                                  | Cross-wind force.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
| $$L$$                                  | Lift force.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| $$g$$                                  | Gravity.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| $$l,m,n$$                              | Moments around coordinate axis x,y and z.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| $$\boldsymbol{\mathrm{M}}$$          | Moment vector $\boldsymbol{\mathrm{M}} = [l \quad m \quad n]^T$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| $$M$$                                  | Mach number. Can be neglected for scale aircraft.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
| $$\boldsymbol{\mathrm{q}}$$          | Vector part of Quaternion.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
| $$\boldsymbol{\mathrm{\tilde{q}}}$$ | Hamiltonian attitude quaternion. $\boldsymbol{\mathrm{\tilde{q}}} = (q_0, q_1, q_2, q_3) = (q_0, \boldsymbol{\mathrm{q}})$.<br> $\boldsymbol{\mathrm{\tilde{q}}}{}$ describes the attitude relative to the local frame $\ell{}$. To represent a vector in local frame given a vector in body frame, the following operation can be used:  $\boldsymbol{\mathrm{\tilde{v}}}^\ell = \boldsymbol{\mathrm{\tilde{q}}} \, \boldsymbol{\mathrm{\tilde{v}}}^b \, \boldsymbol{\mathrm{\tilde{q}}}^*{}$ (or $\boldsymbol{\mathrm{\tilde{q}}}^{-1}{}$ instead of $\boldsymbol{\mathrm{\tilde{q}}}^*{}$ if $\boldsymbol{\mathrm{\tilde{q}}}{}$ is not unitary). $\boldsymbol{\mathrm{\tilde{v}}}{}$ represents a *quaternionized* vector: $\boldsymbol{\mathrm{\tilde{v}}} = (0,\boldsymbol{\mathrm{v}})$ |
| $$\boldsymbol{\mathrm{R}}_\ell^b$$  | Rotation matrix. Rotates a vector from frame $\ell{}$ to frame $b{}$. $$\boldsymbol{\mathrm{v}}^b = \boldsymbol{\mathrm{R}}_\ell^b \boldsymbol{\mathrm{v}}^\ell$$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
| $$\Lambda$$                           | Leading-edge sweep angle.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| $$\lambda$$                           | Taper ratio: $\lambda = c_{tip}/c_{root}$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
| $$w$$                                  | Wind velocity.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
| $$p,q,r$$                              | Angular rates around body axis x,y and z.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| $$\boldsymbol{\omega}^b$$            | Angular rate vector in body frame: $\boldsymbol{\omega}^b = [p \quad q \quad r]^T$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| $$\boldsymbol{\mathrm{x}}$$          | General state vector.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |

### Subscripts / Indices

| Subscripts / Indices | Description                                                      |
| -------------------- | ---------------------------------------------------------------- |
| $$a$$                | Aileron.                                                         |
| $$e$$                | Elevator.                                                        |
| $$r$$                | Rudder.                                                          |
| $$Aero$$             | Aerodynamic.                                                     |
| $$T$$                | Thrust force.                                                    |
| $$w$$                | Relative airspeed.                                               |
| $$x,y,z$$            | Component of vector along coordinate axis x, y and z.            |
| $$N,E,D$$            | Component of vector along global north, east and down direction. |

<a id="superscripts"></a>

### Superscripts / Indices

| Superscripts / Indices | Description                                     |
| ---------------------- | ----------------------------------------------- |
| $$\ell$$              | Local-frame. Default for PX4 related variables. |
| $$b$$                  | Body-frame.                                     |
| $$w$$                  | Wind-frame.                                     |


## Decorators

| Decorator       | Description        |
| --------------- | ------------------ |
| $$()^*$$        | Complex conjugate. |
| $$\dot{()}$$   | Time derivative.   |
| $$\hat{()}$$   | Estimate.          |
| $$\bar{()}$$   | Mean.              |
| $$()^{-1}$$     | Matrix inverse.    |
| $$()^T$$        | Matrix transpose.  |
| $$\tilde{()}$$ | Quaternion.        |

